Log-gamma function

Remarks

Existence: $D$ is simply connected and $\Gamma$ has no zeros on $D$, so $\Gamma$ has a holomorphic logarithm on $D$. This logarithm is unique up to adding $2\pi i k$, and condition 2 fixes $k$. Example: $\operatorname{Log}\Gamma(3) = \ln 2$, which is real, and $e^{\ln 2} = 2 = \Gamma(3)$. Non-example: the composition $z \mapsto \operatorname{Log}(\Gamma(z))$ with the principal logarithm satisfies condition 1 but is not continuous on $D$. The reason is that the argument of $\Gamma(x + iy)$ grows without bound as $y \to \infty$, so $\Gamma(z)$ crosses the negative real axis. The functional equation becomes $\operatorname{Log}\Gamma(z + 1) = \operatorname{Log}\Gamma(z) + \operatorname{Log} z$ on $D$. Its derivative is the digamma function $\psi = \Gamma'/\Gamma$.